TCP: Local stability and Hopf bifurcation

Local Hopf bifurcation and global existence of periodic solutions in TCP system

“…In this paper, we not only consider the local stability and the local Hopf bifurcation of system (3) but also investigate the global Hopf bifurcation and the stability and direction of the local Hopf bifurcation for system (3). Therefore, the present work generalizes and improves the previous results in [6].…”

“…The above analysis leads to the following results: Lemma 1 (i) If a b, then (6) has no root on the imaginary axis for all τ 0.…”

“…In [6], Raina obtained the sufficient condition (i.e., M Cτ > γ (γ − 1)/2 ) that ensures the local stability of system (1) and gave the value of the Hopf bifurcation of system (1). However, he did not investigate the stability and direction of the Hopf bifurcation for system (1).…”

“…It is worth pointing out that a common feature of the work mentioned above is that their analysis is restricted to linear systems and obtained by the linearization of the model equilibrium point. In 2007, Raina [6] analyzed a fluid model of the TCP with an approximation of the drop tail,…”

“…Meanwhile, by choosing an exogenous parameter (not delay τ ) which is denoted by η as the bifurcation parameter, the author also obtained the condition of the existence of the Hopf bifurcation. For more details, one can see [6]. We know that an important characteristic of the TCP is self-clocking, i.e., the sender uses the acknowledgement from the receiver to prompt a step forward.…”